An introduction to ultrametric summability theory PDF
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SPRINGER BRIEFS IN MATHEMATICS P. N. Natarajan An Introduction to Ultrametric Summability Theory 123 SpringerBriefs in Mathematics Series Editors Krishnaswami Alladi Nicola Bellomo Michele Benzi Tatsien Li Matthias Neufang Otmar Scherzer Dierk Schleicher Vladas Sidoravicius Benjamin Steinberg Yuri Tschinkel Loring W. Tu G. George Yin Ping Zhang For furthervolumes: http://www.springer.com/series/10030 SpringerBriefs in Mathematics showcases expositions in all areas of mathe- matics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. This book is published in cooperation with Forum for Interdisciplinary Mathe- matics (FIM). More information on FIM can be found under http://www.forum 4interdisciplinarymathematics.org/. FIM’s Publication Committee BhuDev Sharma (Chair of FIM), Professor, Mathematics, JIIT Noida; P. V. Subrahmanyam (Co-Chair of FIM), Professor, Mathematics, IIT Madras. P. N. Natarajan An Introduction to Ultrametric Summability Theory 123 P. N.Natarajan Retired Professorand Head Department of Mathematics RamakrishnaMission Vivekananda College Chennai, TamilNadu India ISSN 2191-8198 ISSN 2191-8201 (electronic) ISBN 978-81-322-1646-9 ISBN 978-81-322-1647-6 (eBook) DOI 10.1007/978-81-322-1647-6 SpringerNewDelhiHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013948444 (cid:2)TheAuthor(s)2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Dedicated to my parents S. Thailambal and P. S. Narayanasubramanian Preface The purpose of the present monograph is to discuss briefly what summability theory is like when the underlying field is not R (the field of real numbers) or C (the field of complex numbers) but a field K with a non-Archimedean or ultra- metric valuation, i.e., a mapping |(cid:2)|: K !R satisfying the ultrametric inequality jx þ yj (cid:3) maxðjxj;jyjÞ instead of the usual triangle inequality jx þ yj (cid:3) jxjþ jyj; x; y 2 K. Tomakethemonographreallyusefultothosewhowishtotakeupthestudyof ultrametric summability theory and do some original work therein, some knowl- edge of real and complex analysis, functional analysis and summability theory over R or C is assumed. Some of the basic properties of ultrametric fields—their topological structure and geometry—are discussed in Chap. 1. In this chapter, we introduce the p-adic valuation, p being prime and prove that any valuation of Q (the field of rational numbers) is either the trivial valuation, a p-adic valuation or a power ofthe usual absolute value |(cid:2)| on R, i.e.,j(cid:2)ja , where 0\a(cid:3)1. We discuss equivalent valu- ? 1 ations too. In Chap. 2, we discuss some arithmetic and analysis in Q , the p-adic p field for a prime p. In Chap. 2, we also introduce the concepts of differentiability and derivatives in ultrametric analysis and very briefly indicate how ultrametric calculus is different from our usual calculus. In Chap. 3, we speak of ultrametric Banach space, and also mention the many results of the classical Banach space theory, viz., the closed graph, the open mapping and the Banach-Steinhaus theorems carry over in the ultrametric set-up. However, the Hahn-Banach theorem fails to hold. To salvage the Hahn-Banach theorem,theconceptofa‘‘sphericallycompletefield’’isintroducedandIngleton’s version of the Hahn-Banach theorem is proved. The lack of ordering in an ultra- metricfieldKmakesitquitedifficulttofindasubstituteforclassical‘‘convexity’’. However, classical convexity is replaced, in the ultrametric setting, by a notion called ‘‘K-convexity’’, which is briefly discussed towards the end of the chapter. In the main Chap. 4, our survey of the literature on ‘‘Ultrametric Summability Theory’’,startswith thepaper ofAndreeandPetersenof1956(itwastheearliest known paper on the topic) to the present. As far as the author of the present monograph knows, most of the material discussed in the survey has not appeared in book form earlier. Almost all of Chap. 4 consists entirely of the work of the vii viii Preface author of the present monograph. Suitable references have been provided at appropriate places indicating where further developments may be found. The author takes this opportunity to thank Prof. P. V. Subrahmanyam of the DepartmentofMathematics,IndianInstituteofTechnology(Madras),Chennaifor encouraging him to write the monograph during the author’s short stay at the Institute (July 8–August 5, 2011) as a Visiting Faculty. The author thanks the Forum for Inter-disciplinary Mathematics for being instrumental in getting the monograph published. Chennai, India P. N. Natarajan [email protected] Contents 1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Some Arithmetic and Analysis in Q : Derivatives p in Ultrametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Ultrametric Functional Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Ultrametric Summability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Classes of Matrix Transformations. . . . . . . . . . . . . . . . . . . . . 29 4.2 Steinhaus Type Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 Core of a Sequence and Knopp’s Core Theorem . . . . . . . . . . . 44 4.4 A Characterization of Regular and Schur Matrices. . . . . . . . . . 47 4.5 Cauchy Multiplication of Series. . . . . . . . . . . . . . . . . . . . . . . 54 4.6 Nörlund Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.7 Weighted Mean Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.8 Y-method and M-method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.9 Product Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.10 Euler and Taylor Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.11 Tauberian Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.12 Some More Properties of the Nörlund Method . . . . . . . . . . . . 85 4.13 Double Sequences and Double Series. . . . . . . . . . . . . . . . . . . 88 4.14 Nörlund Method for Double Sequences . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 ix Chapter 1 Introduction and Preliminaries Abstract Some basic properties of ultrametric fields—their topological structure andgeometryarediscussedinthischapter.Weintroducethep-adicvaluation,pbeing prime,andprovethatanyvaluationofQ(thefieldofrationalnumbers)iseitherthe trivialvaluation,ap-adicvaluationorapoweroftheusualabsolutevalue,wherethe powerispositiveandlessthanorequalto1.Wediscussequivalentvaluationstoo. · · · Keywords Archimedeanaxiom Ultrametricvaluation Ultrametricfield p-adic · · · valuation p-adicfield p-adicnumbers Equivalentvaluations Thepurposeofthisbookistointroducea“NEWANALYSIS”tostudentsofMath- ematics at the undergraduate and post-graduate levels, which in turn introduces a geometryverydifferentfromthefamiliarEuclideangeometryandRiemanniangeom- etry.Strangethingshappen:forinstance,‘everytriangleisisosceles’and‘everypoint ofasphereisacentreofthesphere’!. ‘Analysis’isthatbranchofMathematicswhereweusetheideaoflimitsexten- sively. A study of Analysis starts with limits, continuity, differentiability, etc., and almostallmathematicalmodelsaregovernedbydifferentialequationsoverthefield Rofrealnumbers.RhasageometrywhichisEuclidean.Imagineapygmytortoise tryingtotravelalongaverylongpath;assumethatitsdestinationisataverylong distance fromitsstartingpoint.Ifatevery step,thepygmy tortoisecovers asmall distance(cid:2),caniteverreachitsdestination,assumingthatthetortoisehasinfinitelife? Ourcommonsensesays“Yes”.ItisoneoftheimportantaxiomsinEuclideangeom- etrythat“Anylargesegmentonastraightlinecanbecoveredbysuccessiveaddition ofsmallsegmentsalongthesameline”.Itisequivalenttothestatement:“givenany number M > 0, there exists an integer N such that N > M”. This is familiarly knownasthe“Archimedeanaxiom”ofthrealnumberfieldR.Whatwouldhappen ifwedonothavethisaxiom?Aretherefieldswhicharenon-Archimedean?Inthe sequel,wewillshowthatsuchfieldsexistandthemetriconsuchfieldsintroduces P.N.Natarajan,AnIntroductiontoUltrametricSummabilityTheory, 1 SpringerBriefsinMathematics,DOI:10.1007/978-81-322-1647-6_1, ©TheAuthor(s)2014
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